The development of abstract algebra brought with itself group theory, rings and fields, Galois theory.

Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born.

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order ( the axiom of commutativity ).

The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

* Alternative algebra, an abstract algebra with alternative multiplication

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.

Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more.

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

Homological algebra is category theory in its aspect of organising and suggesting manipulations in abstract algebra.

In abstract algebra, the derivative is interpreted as a morphism of modules of Kähler differentials.

It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis.

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group.

The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

* Principal ideal domain, in abstract algebra, an integral domain in which every ideal is principal

In computer science, a queue ( ) is a particular kind of abstract data type or collection in which the entities in the collection are kept in order and the principal ( or only ) operations on the collection are the addition of entities to the rear terminal position and removal of entities from the front terminal position.

In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.

The licensed Japanese teacher is normally in charge of junior high school and high school classes regardless of where the ALT originates ; however in the case of elementary school classes, the ALT is normally responsible for the entire class, with the Japanese teacher either providing limited input or in some cases not being present in the classroom, and for that reason the continuity of school management is sometimes maintained with the school principal in compliance with any legal requirements, as the product being contracted itself is an abstract, education, and the contract basis for private language teaching corporations is to provide education and educational services.

A key aspect of the Cartan connection point of view is to elaborate this notion in the context of principal bundles ( which could be called the " general or abstract theory of frames ").

On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstract principal bundle over it.

In abstract algebra, a discrete valuation ring ( DVR ) is a principal ideal domain ( PID ) with exactly one non-zero maximal ideal.

A conference abstract presented in 2009 indicated that long-term nasal irrigation led to higher rates of sinus infections, which the principal researcher theorized was due to alterations in nasal immunological chemistry brought on by flushing out the protective elements of the mucous membrane of the nose.

The input to the DFT is a finite sequence of real or complex numbers ( with more abstract generalizations discussed below ), making the DFT ideal for processing information stored in computers.

Nor do all metaphysical idealists agree on the nature of the ideal ; for Plato, the fundamental entities were non-mental abstract forms, while for Leibniz they were proto-mental and concrete monads.

He showed more zeal and interest in the ascetic ideal than in abstract speculation.

According to Weber's theses, social research cannot be fully inductive or descriptive, because understanding some phenomenon implies that the researcher must go beyond mere description and interpret it ; interpretation requires classification according to abstract " ideal ( pure ) types ".

* Radical of an ideal, an important concept in abstract algebra

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

Peirce defines truth as follows: " Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.

Plato was one of the first essentialists, believing in the concept of ideal forms, an abstract entity of which individual objects are mere facsimilies.

It occurs in the proofs of several theorems of crucial importance, for instance the Hahn – Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.

In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules.

The Prince is sometimes claimed to be one of the first works of modern philosophy, especially modern political philosophy, in which the effective truth is taken to be more important than any abstract ideal.

Renaissance culture did not question the abstract ideal of the Pope as God's human representative on Earth.

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals.

Peirce emphasized fallibilism, considered the assertion of absolute certainty a barrier to inquiry, and in 1901 defined truth as follows: " Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth .".

* Boolean prime ideal theorem, which guarantees the existence of certain types of subsets in a given abstract algebra

Since this is an actual person or fictional character, it is too complex and multi-faceted to be considered an ideal in the abstract sense.

* Emmy Noether publishes Idealtheorie in Ringbereichen, developing ideal ring theory, an important text in the field of abstract algebra.